Euler Characteristics Of Categories And Barycentric Subdivision

We show that three Euler characteristics of categories, Leinster's Euler characteristic, the series Euler characteristic and the Euler characteristic of N-filtered acyclic categories, are invariant under barycentric subdivision for finite acyclic categories. An acyclic category is a small category in which all endomorphisms and isomorphisms are identities, that is, an acyclic category is a skeletal scwol. We show for any small category I, the opposite subdivision category Sd(I)(op) is of type (L-2) if and only if I is finite acyclic. We also extend the definition of the L-2-Euler characteristic and prove our extended L-2-Euler characteristic is invariant under barycentric subdivision for a wider class of finite categories.